Pythagorean theorem

The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple).

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a relation in Euclidean geometry between the three sides of a right triangle. In the West, the theorem is named after the Greek mathematician Pythagoras, who is credited with the first abstract proof.

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the following equation:

$a^{2}+b^{2}=c^{2}.\,$

This equation provides a simple relation between the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them.

This theorem may have a greater variety of known proofs than any other. The Pythagorean Proposition, a book published in 1940, contains 370 different proofs of Pythagoras's theorem, including one by American President James Garfield.

The converse of the theorem is also true:

For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.

History

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC

The history of the theorem can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, and proofs of the theorem.

Megalithic monuments from as early as 4000 BC in Egypt, and in the British Isles from circa 2500 BC, incorporate right triangles with integer sides[1][2] but the builders did not necessarily understand the theorem itself[3][4]. Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically. Written between 2000–1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem, the solution to which is a Pythagorean triple. During the reign of Hammurabi, the Mesopotamian tablet Plimpton 322, written between 1790–1750 BC, contains a large number of entries closely related to Pythagorean triples.

The Baudhayana Sulba Sutra, written in the 8th century BC in India, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India. Many scholars find van der Waerden and Bŭrk's claims unsubstantiated.

Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. Proklos, however, wrote between 410–485 AD. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero did attribute the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.

Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant abstract proof of the theorem is presented.

Written sometime between 500 BC and 200 BC, the Chinese text Chou Pei Suan Ching gives a visual proof of the Pythagorean theorem for the (3, 4, 5) triangle. During the Han Dynasty, from 200 BC to 200 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles.

There has been much debate on whether the Pythagorean theorem was discovered once or many times. B.L. van der Waerden asserts a single discovery, by someone in Neolithic Britain, knowledge of which then spread to Mesopotamia circa 2000 BC, and from there to India, China, and Greece by 600 BC. Most scholars disagree however, and favor independent discovery.

In the West, the theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras. In China, the theorem goes by the name "Gougu Theorem" (勾股定理), based on the numerical proof in the Chou Pei Suan Ching (周髀算经), presented in The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, variously dated between 500 BC–200 AD.

Proofs

This theorem may have a greater variety of known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains over 350 different proofs. James Garfield, who later became a President of the United States, devised an original proof of the Pythagorean theorem in 1876. The external links below provide a sampling of the many different proofs of the Pythagorean theorem.

Some arguments based on trigonometric identities (such as Taylor series for sine and cosine) have been proposed as proofs for the theorem. However, since all the fundamental identities are actually proved using the Pythagorean theorem itself, there cannot be any trigonometric proof. For similar reasons, no proof can be based on analytic geometry or calculus.

More recently, Shri Bharati Krishna Tirthaji in his book[5] claimed ancient Indian Hindu Vedic proofs for the Pythagoras Theorem.

Geometrical proof

Like many of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios:

${\frac {AC}{AB}}={\frac {AH}{AC}}\ {\mbox{and}}\ {\frac {CB}{AB}}={\frac {HB}{CB}}.\,$

These can be written as:

$AC\times AC=AB\times AH$ and $CB\times CB=AB\times HB.$

Summing these two equalities, we obtain:

$AC\times AC+CB\times CB=AB\times AH+AB\times HB=AB\times (AH+HB)=AB\times AB.\,$

In other words, the Pythagorean theorem:

$AC^{2}+BC^{2}=AB^{2}.\,$

Visual proofs

A visual proof is given by the following illustration:

The area of each large square is (a + b)². In both, the area of four identical triangles is removed. The remaining areas, a² + b² and c², are equal. Q.E.D.

This proof is very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is quite easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link).

A second graphic illustration of the Pythagorean theorem fits parts of the sides' squares into the hypotenuse's square. A related proof would show that the repositioned parts are identical with the originals and, since the sum of equals are equal, that the corresponding areas are equal. To show that a square is the result one must show that the length of the new sides equals c.

File:PythagoreanGraphic.jpg

Algebraic proof

File:Pythagproof.png

A square created by aligning four right angle triangles and a large square.

A more algebraic variant of this proof is provided by the following reasoning. Looking at the illustration, the area of each of the four red, yellow, green and pink right-angled triangles is given by:

${\frac {1}{2}}ab$

The blue square in the middle has side length c, so its area is c². Thus the area of the large square is given by:

$4\left({\frac {1}{2}}ab\right)+c^{2}.$

However, as the large square has sides of length a + b, we can also calculate its area as (a + b)(a + b). This can be shown by considering the angles.

Therefore:

$(a+b)(a+b)=4\left({\frac {1}{2}}ab\right)+c^{2}.$

Simplifying this equation:

$a^{2}+b^{2}+2ab=2ab+c^{2},\,$

$a^{2}+b^{2}=c^{2}.\,$

Rational trigonometry

For a proof by the methods of rational trigonometry, see Pythagoras' theorem proof (rational trigonometry).

Converse of the theorem

The converse of the Pythagorean theorem is also true:

For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.

This converse also appears in Euclid's Elements. It can be proven using the law of cosines (see below under Generalizations), or by the following proof:

Let ABC be a triangle with side lengths a, b, and c, with a² + b² = c². We need to prove that the angle between the a and b sides is a right angle. We construct another triangle with a right angle between sides of lengths a and b. By the Pythagorean theorem, it follows that the hypotenuse of this triangle also has length c. Since both triangles have the same side lengths a, b and c, they are congruent, and so they must have the same angles. Therefore, the angle between the side of lengths a and b in our original triangle is a right angle.

Proof by differential equations

One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse in the following diagram and employing a little calculus. This argument appeared in Pythagoras Made Difficult by Michael Hardy in the Mathematical Intelligencer v. 10, no. 3, p. 31, 1988

File:PythagoreanDerivation.GIF

As a result of a change in side a,

${\frac {da}{dc}}={\frac {c}{a}}$

by similar triangles and for differential changes. So

$c\,dc=a\,da$

on rearrangement. A more general result is

$c\ dc=a\,da+b\,db$

which results from adding a second term for changes in side b.

Integrating gives $c^{2}=a^{2}+b^{2}+\mathrm {constant} .\$

$a=b=c=0\Rightarrow \mathrm {constant} =0$

So

$c^{2}=a^{2}+b^{2}.\$

As can be seen, the squares are due to the particular proportion between the changes and the sides while the sum is a result of the independent contributions of the changes in the sides which is not evident from the geometric proofs. From the proportion given it can be shown that the changes in the sides are inversely proportional to the sides. The differential equation suggests that the theorem is due to relative changes and its derivation is nearly equivalent to computing a path integral. A simpler derivation would leave $b\$ fixed and then observe that

$a=0\Rightarrow c^{2}=b^{2}=\mathrm {constant} .$

It is doubtful that the Pythagoreans would have been able to do the above proof but they knew how to compute the area of a triangle and were familiar with figurate numbers and the gnomon, a segment added onto a geometrical figure. All of these ideas are predate calculus and are an alternative for the integral.

Consequences and uses of the theorem

Pythagorean triples

Main article: Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that $a^{2}+b^{2}=c^{2}$. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the British Isles shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5).

A Pythagorean triple is primitive if a, b, and c have no common divisor other than 1. There are infinitely many primitive triples, and all Pythagorean triples can be explicitly generated using the following formula: choose two integers m and n with m > n, and let $a=m^{2}-n^{2}$, $b=2mn$, $c=m^{2}+n^{2}$. Then we have $a^{2}+b^{2}=c^{2}$. Also, any multiple of a Pythagorean triple is again a Pythagorean triple.

Pythagorean triples allow the construction of right angles. The fact that the lengths of the sides are integers means that, for example, tying knots at equal intervals along a string allows the string to be stretched into a triangle with sides of length three, four, and five, in which case the largest angle will be a right angle. This methods was used to step masts at sea and by the Egyptians in construction work.

A generalization of the concept of Pythagorean triples is a triple of positive integers a, b, and c, such that aⁿ + bⁿ = cⁿ, for some n strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's last theorem, though it is doubtful that Fermat ever proved it, and it was far from the last theorem he announced. The first proof was given by Andrew Wiles in 1994.

The existence of irrational numbers

One of the consequences of the Pythagorean theorem is that irrational numbers, such as the square root of two, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of two. The Pythagoreans proved that the square root of two is irrational, and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational. According to the legend, Hippasus, who first proved the irrationality of the square root of two, was drowned at sea as a consequence.

Distance in Cartesian coordinates

The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If (x₀, y₀) and (x₁, y₁) are points in the plane, then the distance between them, also called the Euclidean distance, is given by

${\sqrt {(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}.$

More generally, in Euclidean n-space, the Euclidean distance between two points $X=(x_{1},x_{2},\dots ,x_{n})$ and $Y=(y_{1},y_{2},\dots ,y_{n})$, is defined, using the Pythagorean theorem, as:

${\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+\cdots +(x_{n}-y_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(x_{i}-y_{i})^{2}}}$

Generalizations

The Pythagorean theorem was generalised by Euclid in his Elements:

If one erects similar figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.

The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:

$a^{2}+b^{2}-2ab\cos {\theta }=c^{2},\,$

where θ is the angle between sides a and b.

When θ is 90 degrees, then cos(θ) = 0, so the formula reduces to the usual Pythagorean theorem.

Given two vectors v and w in a complex inner product space, the Pythagorean theorem takes the following form:

$\|\mathbf {v} +\mathbf {w} \|^{2}=\|\mathbf {v} \|^{2}+\|\mathbf {w} \|^{2}+2\,{\mbox{Re}}\,\langle \mathbf {v} ,\mathbf {w} \rangle$

In particular, ||v + w||² = ||v||² + ||w||² if and only if v and w are orthogonal.

Using mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors. Let v₁, v₂,..., v_n be vectors in an inner product space such that <v_i, v_j> = 0 for 1 ≤ i < j ≤ n. Then

$\left\|\,\sum _{k=1}^{n}\mathbf {v} _{k}\,\right\|^{2}=\sum _{k=1}^{n}\|\mathbf {v} _{k}\|^{2}$

The generalisation of this result to infinite-dimensional real inner product spaces is known as Parseval's identity.

When the theorem above about vectors is rewritten in terms of solid geometry, it becomes the following theorem. If lines AB and BC form a right angle at B, and lines BC and CD form a right angle at C, and if CD is perpendicular to the plane containing lines AB and BC, then the sum of the squares of the lengths of AB, BC, and CD is equal to the square of AD. The proof is trivial.

Another generalization of the Pythagorean theorem to three dimensions is de Gua's theorem: If a tetrahedron has a right angle corner (a corner like a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces.

There are also analogs of these theorems in dimensions four and higher.

Edsger Dijkstra discovered this related proposition:

sgn(α + β − γ) = sgn(a² + b² − c²)

where α is the angle opposite to side a, β is the angle opposite to side b and γ is the angle opposite to side c.

This formula holds in all triangles, not just the right triangles which makes it a non-trivial generalization of the Pythagorean theorem. If γ is a right angle (γ equals $\pi /2$ radians or 90°), then sgn(α + β − γ) = 0 since the sum of the angles of a triangle is $\pi$ radians (or 180°). Thus, a² + b² − c² = 0.

In a triangle with three acute angles, α + β > γ holds. Therefore, a² + b² > c² holds.

In a triangle with an obtuse angle, α + β < γ holds. Therefore, a² + b² < c² holds.

The Pythagorean theorem in non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. (It has been shown in fact to be equivalent to Euclid's Parallel (Fifth) Postulate.) For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to $\pi /2$; this violates the Euclidean Pythagorean theorem because $(\pi /2)^{2}+(\pi /2)^{2}\neq (\pi /2)^{2}$.

This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider -- spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:

For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form

$\cos \left({\frac {c}{R}}\right)=\cos \left({\frac {a}{R}}\right)\,\cos \left({\frac {b}{R}}\right).$

By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.

For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form

$\cosh c=\cosh a\,\cosh b$

where cosh is the hyperbolic cosine.

By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.

Other facts

The theorem is referenced in an episode of The Simpsons. After finding a pair of Henry Kissinger's glasses at the Springfield Nuclear Power Plant, Homer puts them on and in an attempt to sound smart, comments "the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (This was a homage to The Wizard of Oz. When the Scarecrow receives his diploma from the Wizard, he recites the Pythagorean theorem incorrectly).

In 2000, Uganda released a coin with the shape of a right triangle. The tail has an image of Pythagoras and the Pythagorean theorem, accompanied with the mention "Pythagoras Millennium".

See also

• Baudhayana
• Katyayana
• linear algebra
• orthogonality
• parallelogram law
• synthetic geometry

References

• John L. Bell, The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development, Kluwer, 1999. ISBN 0792359720.
• Euclid, The Elements, Translated with an introduction and commentary by Sir Thomas L. Heath, Dover, (3 vols.), 2nd edition, 1956.
• Thomas L. Heath. A History of Greek Mathematicians, 2 vols. Clarendon Press, Oxford, 1921. Republished by Dover, New-York, 1981.
• Elisha Scott Loomis The Pythagorean proposition. Washington, D.C : The National Council of Teachers of Mathematics, 1972.
• John Stillwell. Mathematics and Its History, Springer-Verlag, 1989. ISBN 0-387-96981-0 and ISBN 3-540-96981-0.
• B.L. van der Waerden, Geometry and Algebra in Ancient Civilizations, Springer, 1983.

External links

• Weisstein, Eric W. "Pythagorean theorem". MathWorld.
• Dijkstra's generalization
• Some animated proofs of the Pythagorean theorem at Math Is Fun